*2.3.1 Governing equations of fluid flow and heat transfer in pipe*

Due to the turbulent flow in HGHE, an appropriate turbulence model should be selected for calculation. The *k* � *ε* equations are the most widely used in numerical computation and engineering applications. They include:

Turbulent kinetic energy equation:

$$\frac{\partial(\rho k)}{\partial \boldsymbol{\tau}} + \frac{\partial(\rho k u\_i)}{\partial \boldsymbol{\alpha}\_i} = \frac{\partial}{\partial \boldsymbol{\alpha}\_j} \left[ \left( \mu + \frac{\mu\_i}{\sigma\_k} \right) \frac{\partial k}{\partial \boldsymbol{\alpha}\_j} \right] + \boldsymbol{G}\_k + \boldsymbol{G}\_b - \rho \boldsymbol{\varepsilon} - \boldsymbol{Y}\_M + \boldsymbol{S}\_k \tag{1}$$

Dissipation rate equation:

*Computational Fluid Dynamics – Recent Advances, New Perspectives and Applications*

$$\frac{\partial(\rho\varepsilon)}{\partial\tau} + \frac{\partial(\rho\varepsilon u\_i)}{\partial\mathbf{x}\_i} = \frac{\partial}{\partial\mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_i}{\sigma\_\varepsilon} \right) \frac{\partial\varepsilon}{\partial\mathbf{x}\_j} \right] + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbf{G}\_k + \mathbf{C}\_{3\varepsilon} \mathbf{G}\_b) - \mathbf{C}\_{2\varepsilon} \rho \frac{\varepsilon^2}{k} + \mathbf{S}\_\varepsilon \tag{2}$$

where *ρ* is the fluid density, kg/m<sup>3</sup> . *τ* is the time, s. *k*, *ε* respectively represents the dissipation rate of turbulent kinetic energy and pulsation energy. *ui*, *uj* is the hourly average speed, m/s. *μ*, *μ<sup>i</sup>* respectively represent the dynamic viscosity and turbulent viscosity. *Gk* is the turbulent kinetic energy generation term caused by average velocity gradient. *Gb* is the turbulent kinetic energy generation term. *YM* is the contribution of pulsation expansion. *C*1*<sup>ε</sup>*, *C*2*<sup>ε</sup>*, and *C*3*<sup>ε</sup>* are the empirical constants, the values of which are 1.44, 1.92, and 0.09, respectively. *σk*, *σε* are the Pr number, and the values are 1 and 1.3, respectively. *Sk*, *S<sup>ε</sup>* are the user-defined source entries.

The circulating fluid in the buried pipe is water, which is an incompressible fluid and ignores the action of gravity. Thus, it can be simplified as:

Turbulent kinetic energy equation:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_i}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + \mathbf{G}\_k - \rho \varepsilon \tag{3}$$

Dissipation rate equation:

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho\varepsilon u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_i}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \frac{\varepsilon}{k} (\mathbf{C}\_{1\varepsilon} \mathbf{G}\_k - \mathbf{C}\_{2\varepsilon} \rho \varepsilon) \tag{4}$$

Among them, the Prandtl number, turbulent kinetic energy *k*, and dissipation rate *ε* can be automatically calculated by FLUENT software through fluid physical parameters and velocity.

The standard *k* � *ε* model, mass conservation equation, momentum conservation equation, and energy conservation equation together constitute the governing equation of flow and heat transfer, the general form of which is:

$$\frac{\partial(\rho\phi)}{\partial\tau} + \text{div}(\rho U \phi) = \text{div}\left(\Gamma\_{\phi}\text{grad}\phi\right) + \mathbb{S}\_{\phi} \tag{5}$$

where *ϕ* is the universal variable. Γ*<sup>ϕ</sup>* is the generalized diffusion coefficient. *S<sup>ϕ</sup>* is the generalized source term.

The values of parameters in the general equation are shown in **Table** 2:


**Table 2.**

*The value of each parameter in the general equation.*

### *2.3.2 Governing equations of heat transfer of soil*

The heat transfer process of soil around the spiral buried pipe also follows the general governing Eq. (5), and the values of its parameters are shown in **Table 2**.
